Unit: Rational Functions Chapter 9-5: Adding and Subtracting Rational Expressions Essential Question: How do you add and subtract two rational expressions?

Chapter 9-5: Adding and Subtracting Rational Expressions Just like with fractions, terms need to have a common denominator in order for their numerators to be added or subtracted together. After they’re added/subtracted, fractions can be simplified. Sometimes, you will need to find the least common denominator (LCD). To do this, you find the least common multiple of the denominators.

Chapter 9-5: Adding and Subtracting Rational Expressions To find the least common multiple Find the prime factors of each expression Write each prime factor the greatest number of times it appears in either expression Simplify where possible Find the LCM of 4x2 – 36 and 6x2 + 36x + 54 Step 1 4x2 – 36 6x2 + 36x + 54 Step 2 Step 3 = 4(x2 – 9) = (2)(2) (x – 3) (x + 3) = 6(x2 + 6x + 9) = (2) (3) (x + 3)(x + 3) (2) appears twice, (3) appears once, (x – 3) appears once, (x + 3) appears twice (2)(2)(3)(x – 3)(x + 3)(x + 3) = 12(x – 3)(x + 3)2

Chapter 9-5: Adding and Subtracting Rational Expressions Your Turn #1 Find the LCM of 3x2 – 9x - 30 and 6x + 30 Step 1 3x2 – 9x – 30 6x + 30 Step 2 Step 3 = 3(x2 – 3x – 10) = (3) (x – 5) (x + 2) = 6(x + 5) = (2) (3) (x + 5) (2) appears once, (3) appears once, (x - 5) appears once, (x + 5) appears once, (x + 2) appears once (2)(3)(x – 5)(x + 5)(x + 2) = 6(x – 5)(x + 5)(x + 2)

Chapter 9-5: Adding and Subtracting Rational Expressions Your Turn #2 Find the LCM of 5x2 + 15x + 10 and 2x2 – 8 Step 1 5x2 + 15x + 10 2x2 – 8 Step 2 Step 3 = 5(x2 + 3x + 2) = (5) (x + 2) (x + 1) = 2(x2 – 4) = (2) (x + 2) (x – 2) (2) appears once, (5) appears once, (x + 2) appears once, (x + 1) appears once, (x – 2) appears once (2)(5)(x + 2)(x + 1)(x – 2) = 10(x + 2)(x + 1)(x – 2)

Chapter 9-5: Adding and Subtracting Rational Expressions Now that we understand how to find the least common multiple (which is also our least common denominator), let’s use that to add/subtract fractions We multiply each term in the problem by what is missing from the LCD

Chapter 9-5: Adding and Subtracting Rational Expressions Simplify Find the LCM (x + 4)(x + 1) with (3)(x + 1) = (3)(x + 4)(x + 1) Multiply left side (top and bottom) by (3) Multiply right side (top and bottom) by (x + 4)

Chapter 9-5: Adding and Subtracting Rational Expressions Simplify Find the LCM Multiply left side (top and bottom) by Multiply right side (top and bottom) by (x – 6)(x + 2) with (2)(2)(x + 2) = (4)(x – 6)(x + 2) (4) (x - 6)

Chapter 9-5: Adding and Subtracting Rational Expressions Simplify Find the LCM Multiply left side (top and bottom) by Multiply right side (top and bottom) by 5(y + 5)(y – 5) with (3)(y + 5) = (3)(5)(y + 5)(y – 5) (3) 5(y - 5)

Chapter 9-5: Adding and Subtracting Rational Expressions Page 517 – 518 Problems 5 – 21 (odd) Show your work!!!

Unit: Rational Functions Chapter 9-5: Adding and Subtracting Rational Expressions Day 3 Essential Question: How do you add and subtract two rational expressions?

Chapter 9-5: Adding and Subtracting Rational Expressions A complex fraction is a fraction that has a fraction in its numerator, denominator, or both. A few examples: To simplify a complex fraction, multiply all terms by the LCD of all embedded fractions

Chapter 9-5: Adding and Subtracting Rational Expressions Simplify This function has two embedded denominators, x and y, so the LCD of all embedded denominators is xy. Multiply all terms by xy.

Chapter 9-5: Adding and Subtracting Rational Expressions Simplify Multiply all terms by: x

Chapter 9-5: Adding and Subtracting Rational Expressions Simplify Multiply all terms by: 2y

Chapter 9-5: Adding and Subtracting Rational Expressions Simplify Multiply all terms by: x(x + 1)(x – 1)

Chapter 9-5: Adding and Subtracting Rational Expressions Page 517 – 518 Problems 22 – 30 (all) Show your work!!!